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Transactions of the American Mathematical Society

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Intertwining differential operators for $ {\rm Mp}(n,\,{\bf R})$ and $ {\rm SU}(n,\,n)$


Author: Hans Plesner Jakobsen
Journal: Trans. Amer. Math. Soc. 246 (1978), 311-337
MSC: Primary 22E45; Secondary 35L99, 47A15
DOI: https://doi.org/10.1090/S0002-9947-1978-0515541-9
MathSciNet review: 515541
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Abstract: For each of the two series of groups, three series of representations $ {U_n}$, $ {D_n}$, and $ {H_n}(n \in Z)$ are considered. For each series of representations there is a differential operator with the property, that raised to the nth power $ (n > 0)$, it intertwines the representations indexed by $ - n$ and n. The operators are generalizations of the d'Alembertian, the Diracoperator and a combination of the two. Unitarity of subquotients of representations indexed by negative integers is derived from the intertwining relations.


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  • [1] R. Godement, Fonctions automorphes, Séminaire Cartan, Université de Paris, 1957-1958.
  • [2] K. Gross and R. Kunze, Bessel functions and representation theory. Il, J. Functional Analysis 25 (1977), 1-49. MR 0453928 (56:12181)
  • [3] L. Gärding, The solution of Cauchy's problem for two totally hyperbolic linear differential equations by means of Riesz integrals, Ann. of Math. 48 (1947), 785-826. MR 0022648 (9:240a)
  • [4] S. Helgason, Differential geometry and symmetric spaces, Academic Press, New York, 1962. MR 0145455 (26:2986)
  • [5] H. P. Jakobsen and M. Vergne, Wave and Dirac operators, and representations of the conformal group, J. Functional Analysis 24 (1977), 52-106. MR 0439995 (55:12876)
  • [6] H. P. Jakobsen, Conformal harmonic analysis and intertwining differential operators, Thesis, Massachusetts Institute of Technology, 1976.
  • [7] B. Kostant, Verma modules and the existence of quasi-invariant differential operators, Non-Commutative Harmonic Analysis, Lecture Notes in Math., vol. 466, Springer-Verlag, Berlin and New York, 1975. MR 0396853 (53:713)
  • [8] H. Rossi and M. Vergne, Representations of certain solvable Lie groups on Hilbert spaces of holomorphic functions, J. Functional Analysis 13 (1973), 324-389. MR 0407206 (53:10989)
  • [9] -, Analytic continuation of the holomorphic discrete series, Acta Math. 136 (1976), 1-59. MR 0480883 (58:1032)
  • [10] A. Salam and G. Mack, Finite-component field representations of the conformai group, Ann. Physics 53 (1969), 174-202. MR 0245267 (39:6578)
  • [11] I. E. Segal, Mathematical cosmology and extragalactic astronomy, Academic Press, New York, 1976. MR 0496337 (58:14894)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1978-0515541-9
Keywords: Representation, holomorphic function, upper half-plane, reproducing kernel, intertwining differential operator, subquotient, unitarity
Article copyright: © Copyright 1978 American Mathematical Society

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